10 Surprising Mathematical Facts

A Reasoner's Miscellany

Since “10 Mind Blowing Mathematical Equations” is one of my most successful articles to date, I decided to write another math list.

Plenty of things in math are downright uninteresting. Who cares that the area of a circle is πr², or that a negative times a negative is a positive? Why should this interest us at all? Perhaps the answer can be found in the most unexpected results, the counterintuitive facts that have sometimes eluded even the best mathematicians.

1. Birthday Paradox


The birthday paradox says that if there are 23 people in a room, there is a more than 50% chance that two people have the same birthday. It seems counterintuitive because the probability of having a birthday on any particular day is only 1/365.

But the difference relies on the fact that we only need two people to have the same birthday as each other

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Well, This Blog Hasn’t Been Updated in a While

I originally meant for this blog to be an outlet of some topics in math that happened to be a bit too abstract for the general audience. After all, it would have made little sense to post such topics on my main blog, which has a more diverse audience in mind.

This blog went pretty low on my priority list, especially as I got busier this year, but hey, we’re all busy, right? I might be posting sporadically on this blog in 2014, but the main action will be at the main blog (again). Happy Holidays!

Immeasurable Sets in R

The following example was brought up in two different classes that I am taking, within a couple days of each other. It is the classic example of an immeasurable set on the interval [0,1]. Now in math, words like measure and measurable have technical definitions, and lead to bizarre results like the Banach-Tarski paradox. I’m going to give a fairly informal explanation here. Continue reading

The Rules of Paradox Club

Everyone logician knows, “The first rule of Tautology Club is the first rule of Tautology Club.”

Some friends and I made some rules for Paradox Club:

  • The first rule of Paradox Club is not the first rule of Paradox Club.
  • The second rule of Paradox Club is, there is no second rule of Paradox Club.
  • The third rule of Paradox Club is, the fourth rule of Paradox Club must sometimes be followed.
  • The fourth rule of Paradox Club is, the third rule of Paradox Club must never be followed.
  • “Yields falsehood when preceded by its quotation when listed as the fifth rule of Paradox Club,” yields falsehood when preceded by its quotation when listed as the fifth rule of Paradox Club.
  • The sixth rule of Paradox Club cannot be proven to be the sixth rule of Paradox Club.
  • The seventh rule of Paradox Club invalidates precisely the rules of Paradox Club which do not invalidate themselves.
  • The eighth rule of Paradox Club is, do not follow any rules of Paradox Club.

#7: The Visible Grid Point Problem

A link to this article in pdf format: [pdf]

Here is a difficult probability question:

Suppose you are standing on an infinitely large square grid at the point (0,0), and suppose that you can see infinitely far but cannot see through grid points. Given a random grid point z = (x,y), where x and y are integers, what is the chance you can see z?

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